Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2020 AMC 10B Problems/Problem 3"
(Rearranged solutions based on elegance. I think the one-sentence version is the most efficient. Let me know if you disagree.)
Line 7: Line 7:
 
<math>\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\  8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3</math>
 
<math>\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\  8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3</math>
  
−
==Solution 1==
+
==Solution 1 (One Sentence)==
 +
We have <cmath>\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},</cmath> from which <math>w:y=\boxed{\textbf{(E)}\ 16:3}.</math>
  
−
WLOG, let <math>w=4</math> and <math>x=3</math>.
+
~MRENTHUSIASM
−
 
 
−
Since the ratio of <math>z</math> to <math>x</math> is <math>1:6</math>, we can substitute in the value of <math>x</math> to get <math>\frac{z}{3}=\frac{1}{6} \implies z=\frac{1}{2}</math>.
 
−
 
 
−
The ratio of <math>y</math> to <math>z</math> is <math>3:2</math>, so <math>\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}</math>.
 
−
 
 
−
The ratio of <math>w</math> to <math>y</math> is then <math>\frac{4}{\frac{3}{4}}=\frac{16}{3}</math> so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math> ~quacker88
 
  
 
==Solution 2==
 
==Solution 2==
Line 23: Line 18:
 
<math>z:x=1:6=2:12</math>, and since <math>y:z=3:2</math>, we can link them together to get <math>y:z:x=3:2:12</math>.
 
<math>z:x=1:6=2:12</math>, and since <math>y:z=3:2</math>, we can link them together to get <math>y:z:x=3:2:12</math>.
  
−
Finally, since <math>x:w=3:4=12:16</math>, we can link this again to get: <math>y:z:x:w=3:2:12:16</math>, so <math>w:y = \boxed{\textbf{(E)}\ 16:3}</math> ~quacker88
+
Finally, since <math>x:w=3:4=12:16</math>, we can link this again to get: <math>y:z:x:w=3:2:12:16</math>, so <math>w:y = \boxed{\textbf{(E)}\ 16:3}</math>.
 +
 
 +
~quacker88
  
 
==Solution 3==
 
==Solution 3==
−
 
 
We have the equations <math>\frac{w}{x}=\frac{4}{3}</math>, <math>\frac{y}{z}=\frac{3}{2}</math>, and <math>\frac{z}{x}=\frac{1}{6}</math>.
 
We have the equations <math>\frac{w}{x}=\frac{4}{3}</math>, <math>\frac{y}{z}=\frac{3}{2}</math>, and <math>\frac{z}{x}=\frac{1}{6}</math>.
 
Clearing denominators, we have <math>3w = 4x</math>, <math>2y = 3z</math>, and <math>6z = x</math>.  
 
Clearing denominators, we have <math>3w = 4x</math>, <math>2y = 3z</math>, and <math>6z = x</math>.  
Line 33: Line 29:
 
to get <math>4y = x</math> . Now, to be able to substitute this into out first equation, we need to have <math>4x</math> on the RHS.  
 
to get <math>4y = x</math> . Now, to be able to substitute this into out first equation, we need to have <math>4x</math> on the RHS.  
 
Multiplying both sides by <math>4</math>, we have <math>16y = 4x</math>.  
 
Multiplying both sides by <math>4</math>, we have <math>16y = 4x</math>.  
−
Substituting this into our first equation, we have <math>3w = 16y</math>, or <math>\frac{w}{y}=\frac{16}{3}</math>, so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math>  
+
Substituting this into our first equation, we have <math>3w = 16y</math>, or <math>\frac{w}{y}=\frac{16}{3}</math>, so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math>.
 +
 
 
~Binderclips1
 
~Binderclips1
  
−
==Solution 4 (One Sentence)==
+
==Solution 4==
−
We have <cmath>\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},</cmath> from which <math>w:y=\boxed{\textbf{(E)}\ 16:3}.</math>
+
 
 +
WLOG, let <math>w=4</math> and <math>x=3</math>.
 +
 
 +
Since the ratio of <math>z</math> to <math>x</math> is <math>1:6</math>, we can substitute in the value of <math>x</math> to get <math>\frac{z}{3}=\frac{1}{6} \implies z=\frac{1}{2}</math>.
 +
 
 +
The ratio of <math>y</math> to <math>z</math> is <math>3:2</math>, so <math>\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}</math>.
 +
 
 +
The ratio of <math>w</math> to <math>y</math> is then <math>\frac{4}{\frac{3}{4}}=\frac{16}{3}</math> so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math>.
  
−
~MRENTHUSIASM
+
~quacker88
  
 
==Video Solution==
 
==Video Solution==

Revision as of 02:16, 26 August 2021

The following problem is from both the 2020 AMC 10B #3 and 2020 AMC 12B #3, so both problems redirect to this page.

Problem

The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$

$\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\ 8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3$

Solution 1 (One Sentence)

We have \[\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},\] from which $w:y=\boxed{\textbf{(E)}\ 16:3}.$

~MRENTHUSIASM

Solution 2

We need to somehow link all three of the ratios together. We can start by connecting the last two ratios together by multiplying the last ratio by two.

$z:x=1:6=2:12$, and since $y:z=3:2$, we can link them together to get $y:z:x=3:2:12$.

Finally, since $x:w=3:4=12:16$, we can link this again to get: $y:z:x:w=3:2:12:16$, so $w:y = \boxed{\textbf{(E)}\ 16:3}$.

~quacker88

Solution 3

We have the equations $\frac{w}{x}=\frac{4}{3}$, $\frac{y}{z}=\frac{3}{2}$, and $\frac{z}{x}=\frac{1}{6}$. Clearing denominators, we have $3w = 4x$, $2y = 3z$, and $6z = x$. Since we want $\frac{w}{y}$, we look to find $y$ in terms of $x$ since we know the relationship between $x$ and $w$. We begin by multiplying both sides of $2y = 3z$ by two, obtaining $4y = 6z$. We then substitute that into $6z = x$ to get $4y = x$ . Now, to be able to substitute this into out first equation, we need to have $4x$ on the RHS. Multiplying both sides by $4$, we have $16y = 4x$. Substituting this into our first equation, we have $3w = 16y$, or $\frac{w}{y}=\frac{16}{3}$, so our answer is $\boxed{\textbf{(E)}\ 16:3}$.

~Binderclips1

Solution 4

WLOG, let $w=4$ and $x=3$.

Since the ratio of $z$ to $x$ is $1:6$, we can substitute in the value of $x$ to get $\frac{z}{3}=\frac{1}{6} \implies z=\frac{1}{2}$.

The ratio of $y$ to $z$ is $3:2$, so $\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}$.

The ratio of $w$ to $y$ is then $\frac{4}{\frac{3}{4}}=\frac{16}{3}$ so our answer is $\boxed{\textbf{(E)}\ 16:3}$.

~quacker88

Video Solution

https://www.youtube.com/watch?v=QqkNnsNEgiA

-Education, the Study of Everything (check it out! *Simple Solution)

https://youtu.be/Gkm5rU5MlOU (for AMC 10)

https://youtu.be/WfTty8Fe5Fo (for AMC 12)

~IceMatrix

https://youtu.be/y4BbRapJepY

~savannahsolver

https://youtu.be/wH7xhYxwaFc

~AlexExplains

See Also

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_AMC_10B_Problems/Problem_3&oldid=160962"